Relativistic Corrections to the Sunyaev‐Zeldovich Effect for Clusters of Galaxies. II. Inclusion of Peculiar Velocities

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Abstract
We extend the formalism of the relativistic thermal Sunyaev-Zeldovich effect to a system moving with a velocity β ≡ /c with respect to the cosmic microwave background radiation. In the present formalism, the kinematic Sunyaev-Zeldovich effect for the cluster of galaxies with a peculiar velocity β is derived in a straightforward manner by the Lorentz boost of the generalized Kompaneets equation. We give an analytic expression for the kinematic Sunyaev-Zeldovich effect, which is valid up to O(β2) with the power series expansion approximation in terms of θe ≡ kBTe/mc2, where Te and m are the electron temperature and the electron mass, respectively. It is found that the relativistic corrections to the kinematic Sunyaev-Zeldovich effect are significant. For a typical electron temperature kBTe = 10 keV, one obtains -8.2% and +1.3% corrections from the O(βθe) and O(βθ2e) contributions, respectively. The O(β2) correction is extremely small: +0.2% for β = 1/300 at kBTe = 10 keV. Therefore it can be safely neglected. These relativistic corrections are directly reflected on the determination of the peculiar velocity β of the cluster of galaxies with the observation of the kinematic Sunyaev-Zeldovich effect.